Analytical Geometry, ODE, PDE, Complex Analysis & Numerical Methods

Analytic Geometry

Cartesian and polar coordinates in three dimensions, second degree equations in three variables, reduction to canonical forms, straight lines, shortest distance between two skew lines; Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

Ordinary Differential Equations

Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory; Equations of first order but not of first degree, Clairaut's equation, singular solution. Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution. Second order linear equations with variable coefficients, Euler-Cauchy equation; Determination of complete solution when one solution is known using method of variation of parameters. Laplace and Inverse Laplace transforms and their properties; Application to initial value problems for 2nd order linear equations with constant coefficients.

Partial Differential Equations

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchy's method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of a vibrating string, heat equation, Laplace equation and their solutions.

Numerical Methods & Computer Programming

Numerical Methods: Bisection, Regula-Falsi and Newton-Raphson methods; Solution of systems via Gaussian elimination and Gauss-Jordan; Gauss-Seidel iterative method. Interpolation: Newton's, Lagrange's methods. Integration: Trapezoidal, Simpson's, Gaussian quadrature. ODE Solvers: Euler, Runge-Kutta.

Computer Programming: Binary/Hexadecimal systems, Boolean algebra, basic gates. Data representation: signed/unsigned integers and reals. Algorithms & flowcharts for numerical problems.